I do not understand a remark in Aronszajn's paper. He said the unique continuation can be also applied to the Monge-Ampère
$$
\det D^2u = F(x, u ,Du)
$$
Edit: It eluded me that Aronszajn assumed the solution to be $\underline{\text{convex}}$ and $C^{4,1}$ and this regularity assumption is to let his theorem applicable.
Also, $F$ is assumed to be Lipschitz except possibly $x$.
However the unique continuation is stated in this form:
If $u$ satisfies for some linear 2nd order elliptic operator $A$ near $0$: $$\vert A u\vert^2\leq M(\vert Du\vert^2 + \vert u \vert^2)$$ and $u$ has zero of infinity order in 1-mean at $0$: $$\int_{\vert x \vert\leq r} \vert u\vert = O(r^{\alpha + n}) \quad \text{for all }\alpha > 0$$ then $u$ is identically zero in that vicinity.
How is the unique continuation applied to Monge-Ampère PDE? I don't know how to find a suitable linear operator $A$.
Sketch: Similar to [Thm 17.1 Gilbarg-Trudinger] let $u^\theta = \theta u + (1-\theta) v$ for two convex solutions and plug in Monge-Ampère, then for $w = u-v$ $$\left(\int^1_0 D^2_{ik} u^\theta \det(D^2 u^\theta) d\theta \right) D^2_{ik} w + b^j \partial_j w + cw = 0$$ Now the foregoing coefficients of $D^2 w$ is elliptic since both $u$ and $v$ are convex and of $C^{2,1}$ regularity. Also note that both $b$ and $c$ make sense in this derivation since $F$ is Lipschitz so in particular Fundamental theorem of Calculus is applicable. Finally, we may apply Aronszajn's result to conclude $w=0$.